Design of quasi-phasematched optical frequency converters

ABSTRACT

Quasi-phasematching design to provide an approximation to a desired spectral amplitude response A(f) is provided. An initial phase response φ(f) corresponding to A(f) is generated. Preferably, d 2 φ(f)/df 2  is proportional to A 2 (f). A function h(x) is computed such that h(x) and H(f)=A(f)exp(iφ(f)) are a Fourier transform pair. A domain pattern function d(x) is computed by binarizing h(x) (i.e., approximating h(x) with a constant-amplitude approximation). In some cases, the response provided by this d(x) is sufficiently close to A(f) that no further design work is necessary. In other cases, the design can be iteratively improved by modifying φ(f) responsive to a difference between the desired response A(f) and the response provided by domain pattern d(x). Various approaches for binarization are provided. The availability of multiple binarization approaches is helpful for making design trades (e.g., in one example, fidelity to A(f) can be decreased to increase efficiency and to increase domain size).

CROSS REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. provisional application60/639,121, filed on Dec. 22, 2004, entitled “Design ofQuasi-phasematched Optical Frequency Converters” and hereby incorporatedby reference in its entirety.

FIELD OF THE INVENTION

This invention relates to nonlinear frequency conversion.

BACKGROUND

Nonlinear optical frequency conversion includes any process where one ormore optical inputs provided to a nonlinear optical device produce oneor more optical outputs, where the output radiation includes radiationat one or more frequencies (or wavelengths) not present in the inputradiation. Examples of nonlinear frequency conversion include secondharmonic generation (SHG), sum frequency generation (SFG), differencefrequency generation (DFG), four wave mixing, third harmonic generation,parametric oscillation, etc. Many nonlinear optical processes requirephase-matching to proceed efficiently. If the phase matching conditionis satisfied, then the nonlinear interaction proceeds constructivelyalong the entire active length of the device, while if the phasematching condition is not satisfied, then radiation from different partsof the nonlinear device interferes destructively to reduce conversionefficiency. As a result, investigation of such processes (e.g., secondorder processes such as DFG, SFG, and SHG) has concentrated primarily onmethods for phase-matching.

The phase-matching condition can be expressed in geometrical terms. Forexample, phase-matching for SHG requires the wave vector of the secondharmonic wave to be twice as long as the wave vector of the input wave(i.e., the pump wave). Due to material dispersion (i.e., the wavelengthdependence of the index of refraction), the SHG phase matching conditionis ordinarily not satisfied. Birefringent phase-matching (BPM) and quasiphase-matching (QPM) are two methods of phase-matching that have beenextensively investigated. In BPM, birefringent materials are employedand the interaction geometry and wave polarization are selected suchthat the phase matching condition is satisfied. For example, the pumpand second harmonic waves can have the same index of refraction tophase-match SHG. In QPM, the nonlinear device is spatially modulated toprovide phase matching. For example, periodic spatial modulation of anonlinear device can provide a device k-vector K such that 2k_(p)±K=k_(sh) to phase-match SHG, where k_(p) is the pump wave vectorand k_(sh) is the second harmonic wave vector. A common method ofproviding spatial modulation for QPM is to controllably alter the signof the nonlinear coefficient (e.g., by poling a ferroelectric material).

In more general terms, QPM can be regarded as a method for engineeringthe spectral response of a nonlinear optical device to provide variousdesirable results. From this point of view, periodic QPM is a specialcase of QPM that is especially appropriate for maximizing conversionefficiency at a single set of input and output wavelengths. Other designconstraints can lead to various non-periodic QPM methods. For example,in U.S. Pat. No. 5,815,307 and U.S. Pat. No. 5,867,304 aperiodic QPM isemployed in connection with frequency conversion of short opticalpulses. Since short pulses include multiple wavelengths, periodic QPMoptimized for a single wavelength is not preferred. In U.S. Pat. No.6,016,214, a QPM grating having multiple sections, each having adifferent period, is employed to phase-match multiple nonlinearprocesses. In U.S. Pat. No. 6,714,569 and U.S. Pat. No. 5,640,405, QPMfor two or more nonlinear processes simultaneously is also considered.

Increasing the wavelength acceptance bandwidth by periodic or aperiodicphase reversal is considered by Bortz et al., in Electronic Letters30(1), pp34–35, 1994. Sinusoidally chirped QPM is considered by Gao etal., in Photonics Technology Letters, 16(2), pp557–559, 2004.Combination of a phase reversal grating and a periodic grating for QPMis considered by Chou et al., in Optics Letters, 24(16), pp1157–1159,1999.

In some cases, it is desirable to specify a nonlinear device spectralresponse (e.g., normalized SHG efficiency) over a range of frequencies.In such cases, the above-described methods may or may not be applicable,depending on whether or not the desired spectral response falls into theset of spectral responses provided by the method. For example, QPMgratings having several periodic sections provide a spectral responsehaving several peaks, each peak having characteristic side lobes due tothe sin(x)/x (i.e., sinc(x)) response from each grating section. If thedesired spectral response is one or several sinc-like peaks, then thismethod is applicable. If the desired spectral response is different(e.g., the side lobes need to be eliminated), then this method may notbe applicable.

Accordingly, it would be an advance in the art to provide QPM having aspecified spectral response (or tuning curve).

SUMMARY

The invention enables QPM design to provide an approximation to adesired spectral amplitude response A(f). An initial phase response φ(f)corresponding to A(f) is generated. Preferably, d²φ(f)/df² isproportional to A²(f). A function h(x) is computed such that h(x) andH(f)=A(f)exp(iφ(f)) are a Fourier transform pair. A domain patternfunction d(x) is computed by binarizing h(x) (i.e., approximating h(x)with a constant-amplitude approximation). In some cases, the responseprovided by this d(x) is sufficiently close to A(f) that no furtherdesign work is necessary. In other cases, the design can be iterativelyimproved by modifying φ(f) responsive to a difference between thedesired response A(f) and the response provided by domain pattern d(x).Various approaches for binarization are provided. The availability ofmultiple binarization approaches is helpful for making design trades(e.g., in one example, fidelity to A(f) can be decreased to increaseefficiency and to increase domain size).

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a QPM method according to an embodiment of the invention.

FIG. 2 shows an example of a spectral conversion efficiencyspecification.

FIGS. 3 a–c show a first example of QPM design according to anembodiment of the invention.

FIGS. 4 a–c show a second example of QPM design according to anembodiment of the invention.

DETAILED DESCRIPTION

FIG. 1 shows a QPM method according to an embodiment of the invention.Step 102 on FIG. 1 is specifying a target spectral amplitude responseA(f). The response A(f) describes the desired spectral (i.e., as afunction of wavelength) performance of the QPM device being designed.More specifically, A²(f) is the desired relative spectral powerefficiency of the device, so A(f) is referred to as an amplituderesponse. In practice, A²(f) may be given, from which A(f) can readilybe derived. Thus the QPM design problem can be regarded as choosing adomain pattern function d(x) that provides a spectral amplitude responseB(f) that is sufficiently close to the target spectral amplituderesponse A(f). The specification of A(f) should span the frequencyrange(s) of interest, including any range(s) where zero response isdesired.

Here it is assumed that QPM is accomplished by changing the sign of thenonlinear coefficient in a controlled manner (e.g., by controlled domainpoling of a ferroelectric nonlinear material). Under this assumption,the domain pattern function d(x) has sign reversals at each domainboundary, but |d(x)| is a constant. It is convenient (but not required)to normalize the calculations such that |d(x)|=1. With thisnormalization of d(x), it is clear that the amplitude response A(f) canbe correspondingly normalized. Thus the response A(f) is to be regardedas a relative response, in the sense that multiplication of A(f) by awavelength-independent factor does not essentially change the QPM designproblem being specified. Such normalization is well known in the art, asis the equivalence between normalized and non-normalized QPM formalisms.

It is also well known in the art that there is a Fourier transformrelation between the domain pattern function d(x) and spectral responseD(f) of a QPM nonlinear device. More specifically, d(x) and D(f) are aFourier transform pair. Here (and throughout this description) x isposition and f is frequency or any suitable equivalent such as spatialfrequency or angular frequency. For example, a wavelength of 1 μmcorresponds to a spatial frequency of 1 μm⁻¹ and to a frequency of 300THz. Since various conventions exist for defining Fourier transforms, nosignificance attaches to convention-dependent details such as thelocation of factors of 2π, sign conventions, the use of one-sided ortwo-sided transforms, and whether D(f) is regarded as the Fouriertransform or inverse Fourier transform of d(x). Any mutually consistentFourier transform formalism will suffice for practicing the invention.

Practice of the invention also does not depend on whether these Fouriertransform pairs are continuous or discrete. In most cases, including thefollowing examples, discrete Fourier transforms are preferred to makeuse of efficient algorithms such as the fast Fourier transform.

Even though there is a simple relation between d(x) and D(f), it doesnot follow that d(x) can readily be derived from the desired responseA(f). To appreciate this, it is helpful to define the amplitude (B(f))and phase (ψ(f)) responses of a QPM device via D(f)=B(f)exp(iψ(f)),where B(f) is real and non-negative and ψ(f) is real. Now it is clearthat specifying A(f) is only a partial specification of the desiredspectral response, since the phase is left unspecified. We assume thatthe designer does not care directly about the phase response (e.g., thecommon case where efficiency B²(f) is the quantity of concern).

However, it does not follow that the phase response is irrelevant to QPMdesign. In fact, a good choice of phase response can improve QPM design(i.e., make B(f) closer to A(f)) and a poor choice of phase response candegrade QPM design (i.e., make B(f) farther from A(f)), other thingsbeing equal. The reason for this is that d(x) is constrained to have aconstant amplitude, and varying the phase response affects the influencethis constraint has on design fidelity (i.e., how closely B(f) canapproach A(f)).

Accordingly, step 104 on FIG. 1 is providing an initial phase responseφ(f). The phase response φ(f) can be regarded as a target phase responsespecification selected to provide good fidelity between A(f) and B(f).Preferably, φ(f) is selected such that d²φ(f)/df² is proportional toA²(f) and more preferably, φ(f)∝∫₀ ^(f)(∫₀ ^(z) ² A²(z₁)dz₁)dz₂. Thischoice of phase response works well in practice, and often provides anon-iterative design solution. The proportionality constant ispreferably set to minimize the standard deviation of the values of h²(x)divided by the average value of h²(x). Thus, the proportionalityconstant is set to make h(x) as uniform as possible given otherconstraints. Standard numerical methods are applicable for determiningthis constant of proportionality, and since this is a one-dimensionaloptimization problem, it is not computationally intensive. Other initialtarget phase responses can also be employed, e.g., in connection withiterative design methods as described below. Specification of the phaseresponse as described herein is a key aspect of the invention.

In addition to having a constant amplitude, d(x) is a real-valuedfunction. Therefore, the responses A(f) and φ(f) are required to havethe corresponding Fourier symmetries (i.e., A(f)=A(−f) and φ(f)=−φ(−f)).Step 106 on FIG. 1 is calculating h(x) such that h(x) andH(f)=A(f)exp(iφ(f)) are a Fourier transform pair. The symmetryconditions on A(f) and φ(f) ensure that h(x) is a real-valued function.If the domain pattern function d(x) could be set equal to (orproportional to) h(x), the QPM design problem would be solved. However,h(x) generally does not have a constant amplitude, so the domain patternfunction d(x) needs to be a constant-amplitude approximation to h(x).

It is helpful to regard the process of deriving a constant-amplitudeapproximation to h(x) as “binarizing” h(x). Accordingly, step 108 onFIG. 1 is binarizing h(x) to provide d(x). Such binarization can beaccomplished in various ways.

One method of binarization is to set d(x) proportional to sgn(h(x)),which effectively looks at only the sign of h(x). Other binarizationmethods that have been developed can be grouped into two categories. Inthe first category, d(x) is set proportional to sgn(h(x)−m(x)), wherem(x) is a modulating waveform. In the second category, d(x) is setproportional to sgn(g(x)−m(x)), where m(x) is a modulating waveform andg(x) is an average of h(x). In addition to these two categories, it isalso possible to include frequency domain processing in the binarizationprocess.

In the first category, d(x) is set proportional to sgn(h(x)−m(x)), wherem(x) is a modulating waveform preferably selected to provide a D(f) thatis substantially proportional to H(f) over a predetermined range of f.Selection of the range of f is problem-dependent and within the skill ofan average art worker. Achieving this condition provides good fidelitybetween A(f) and B(f). Suitable modulating waveforms include trianglewaves and sawtooth waves. A triangle wave has equal positive andnegative slopes in each period and has no discontinuities, while asawtooth wave has only one slope in each period and has discontinuities.Triangle wave modulation is preferred to sawtooth wave modulationbecause the resulting domain sizes tend to be slightly larger. In caseswhere m(x) is periodic with period T, it is preferred for 1/T to begreater than about three times the largest frequency of interestspecified by A(f). The amplitude of m(x) is usually set equal to thepeak amplitude of h(x), although other choices are possible for theamplitude. Random or pseudorandom waveforms for m(x) are also suitablefor practicing the invention. In this description, pseudo-random refersto a deterministic output (e.g., from a random number generator) thatpasses statistical tests for randomness. Spectral filtering of a randomor pseudo-random input can be employed to provide a filtered random orpseudo-random m(x). In practice, good results have been obtained with apseudo-random m(x) having an amplitude uniformly distributed within apredetermined range.

The second category is like the first, except that h(x) is averaged toprovide g(x) inside the sgn( ) function. This average can be a movingaverage. Alternatively, in cases where m(x) is periodic with period T,g(x) can be given by

${g\mspace{11mu}(x)} = {\frac{1}{T}{\int_{{\lfloor{x/T}\rfloor}\mspace{11mu} T}^{{({{\lfloor{x/T}\rfloor} + 1})}\mspace{11mu} T}{h\mspace{11mu}(y)\ {{\mathbb{d}y}.}}}}$Here └z┘ (i.e., floor(z)) is the largest integer≦z. Thus within eachperiod i of m(x), an average value of h_(i) of h(x) is computed. Thevalue of g(x) is the average h_(i) corresponding to the periodcontaining x. Averaging h(x) to g(x) desirably tends to increase domainsize, but in turn tightens tolerances on duty cycle control. To maximizedomain size in cases where m(x) is periodic with period T and averagingof h(x) to g(x) is performed, it is preferred for 1/T to be greater thanabout two times the largest frequency of interest specified by A(f). Theselection of what kind of averaging to employ (if any) can be made by anart worker based on overall design and process considerations, takinginto account the minimum domain size limit and duty cycle tolerances ofthe available domain patterning processes.

Binarization can also include frequency-domain processing. Morespecifically, the D(f) corresponding to a domain pattern function d(x)can be processed to provide a modified function F(f). A function f′(x)is computed such that f′(x) and F(f) are a Fourier transform pair, andf′(x) is binarized to provide f(x) (with any of the above binarizationmethods). Such processing can provide various useful enhancements tobinarization. For example, low pass filtering of D(f) to provide F(f)increases the domain size in f(x) compared to d(x). Increasing thedomain size (especially the minimum domain size) is helpful forfabrication, since smaller domains tend to be more difficult tofabricate than larger domains. The response D(f) can also be filtered toreject QPM harmonics. Spectral processing in combination withbinarization can be an iterative process or a single-pass process. In aniterative process, the processing of D(f) to obtain F(f), thecalculation of f′(x), and the binarization of f′(x) to obtain f(x) arerepeated in sequence until a termination condition is satisfied.

In performing iterative binarization, it has been found helpful tomonitor the convergence of the process. If a condition of slowconvergence is detected, the iteration can be restarted with differentinitial conditions. Iteration after such restarting is found to convergequickly in most cases, making such restarting a surprisingly effectivestrategy. In cases where m(x) is random or pseudorandom, the restartingis performed such that a different m(x) is generated. In other cases(e.g., if m(x) is a triangle wave or sawtooth wave) the phase and/oramplitude of m(x) can be perturbed (deterministically or randomly) aspart of the restarting, to avoid encountering the same convergencedifficulty. Here the phase of m(x) refers to the relative phase betweenm(x) and h(x).

FIG. 1 also shows optional steps 110, 112, and 114. Optional step 110 iscomputation of the response D(f)=B(f)exp(iψ(f)) provided by the domainpattern function d(x). Since the goal is to make B(f) as close aspossible to A(f), standard numerical methods are applicable. A figure ofmerit depending on A(f) and B(f) can be selected (e.g., an integral ofthe square of the difference between A(f) and B(f) in a predeterminedrange or ranges of f)). The input phase response φ(f) (typically in adiscretized approximation) can be varied to improve the figure of merit(step 112). Such variation can be iterated (step 114) according to knownnumerical optimization techniques.

FIG. 2 shows an example of a spectral conversion efficiencyspecification. The target response of FIG. 2 is used in the examples ofFIGS. 3 a–c and 4 a–c. FIG. 2 shows normalized conversion efficiency, sothe corresponding target amplitude response A(f) is the square root ofthe plot of FIG. 2.

FIGS. 3 a–c show a first example of QPM design according to anembodiment of the invention. FIG. 3 a shows the spectral response (i.e.,B²(f)) plotted between 0.5 dB limits computed from the target spectralresponse of FIG. 2. Excellent response fidelity is observed in thisexample. FIG. 3 b shows a section of the domain pattern function d(x) ina 50 μm window 1 mm into the device, and FIG. 3 c shows the spectralresponse over a larger range of spatial frequencies.Binarization-induced peaks (near 0.5 μm⁻¹, 1.7 μm⁻¹ and 2.8 μm⁻¹) arevisible, as are images of the tuning curve. In this example, the phaseresponse is proportional to ∫₀ ^(f)(∫₀ ^(z) ² A²(z₁)dz₁)dz₂, theproportionality constant is optimized as described above, andsgn(h(x)−m(x)) binarization is employed where m(x) is a triangle wavehaving a fundamental frequency of 0.57 μm⁻¹ (3 times the largestspecified frequency in A(f)). The amplitude of m(x) is set equal to thepeak amplitude of h(x).

FIGS. 4 a–c show a second example of QPM design according to anembodiment of the invention. In this example, a trade is made toincrease efficiency and domain size at the expense of somewhat reducedresponse fidelity. This example is the same as the example of FIGS. 3a–c, except that the amplitude of m(x) is decreased by a factor of 3π/4.FIG. 4 a shows the spectral response (i.e., B²(f)). FIG. 4 b shows asection of the domain pattern function d(x) in a 50 μm window 1 mm intothe device, and FIG. 4 c shows the spectral response over a larger rangeof spatial frequencies. Comparison of FIGS. 4 b and 3 b show far fewershort domains in FIG. 4 b, which can greatly facilitate fabrication.Correspondingly, the high spatial frequency content on FIG. 4 c is muchless than on FIG. 3 c. Considerable design flexibility is provided bythe various binarization approaches, and selection of suitablebinarization approaches for particular designs in accordance withprinciples of the invention is within the skill of an art worker.

1. A method for designing a quasi-phasematched optical frequencyconverter, the method comprising: a) specifying a non-negative realtarget spectral amplitude response A(f), wherein A(−f)=A(f); b)providing a real initial phase response φ(f) associated with A(f),wherein φ(f)=−φ(−f); c) calculating a real function h(x) such that h(x)and H(f)=A(f) exp(iφ(f)) are a Fourier transform pair; d) binarizingh(x) to provide a domain pattern function d(x), wherein d(x) is a realconstant-amplitude approximation to h(x).
 2. The method of claim 1,wherein said providing a phase response comprises providing φ(f) suchthat d²φ(f)/df² is proportional to A²(f).
 3. The method of claim 2,wherein said providing a phase response comprises providing φ(f)∝∫₀^(f)(∫₀ ^(z) ² A²(z₁)dz₁)dz₂.
 4. The method of claim 3, wherein φ(f)=K∫₀^(f)(∫₀ ^(z) ² A²(z₁)dz₁)dz₂ and wherein K is selected to minimize astandard deviation of h²(x) divided by an average value of h²(x).
 5. Themethod of claim 1, wherein h(x) and H(f) are a discrete Fouriertransform pair.
 6. The method of claim 1, wherein h(x) and H(f) are acontinuous Fourier transform pair.
 7. The method of claim 1, whereinsaid binarizing comprises setting d(x) substantially proportional tosgn(h(x)).
 8. The method of claim 1 further comprising: e) calculating afunction D(f)=B(f) exp(iψ(f)) such that D(f) and d(x) are a Fouriertransform pair, wherein B(f) is real and non-negative and wherein ψ(f)is real.
 9. The method of claim 8, wherein said binarizing comprisessetting d(x) substantially proportional to sgn(g(x)−m(x)), wherein m(x)is a modulating waveform and wherein g(x) is an average of h(x).
 10. Themethod of claim 9, wherein m(x) is periodic with period T, and wherein${g\mspace{11mu}(x)} = {\frac{1}{T}{\int_{{\lfloor{x/T}\rfloor}\mspace{11mu} T}^{{({{\lfloor{x/T}\rfloor} + 1})}\mspace{11mu} T}{h\mspace{11mu}(y)\ {{\mathbb{d}y}.}}}}$11. The method of claim 9, wherein g(x) is a moving average of h(x). 12.The method of claim 8, wherein said binarizing comprises setting d(x)substantially proportional to sgn(h(x)−m(x)), wherein m(x) is amodulating waveform selected such that D(f) is substantiallyproportional to H(f) over a predetermined range of f.
 13. The method ofclaim 12, wherein said modulating waveform is a triangle wave or asawtooth wave.
 14. The method of claim 12, wherein said modulatingwaveform is random or pseudorandom.
 15. The method of claim 14, whereina power spectral density of said modulating waveform is determined bypassing said waveform through a filter.
 16. The method of claim 14,wherein said modulating waveform has an amplitude that is uniformlydistributed within a predetermined range.
 17. The method of claim 8,wherein said binarizing further comprises: processing said function D(f)to provide a modified function F(f); computing a function f′(x) suchthat f′(x) and F(f) are a Fourier transform pair; binarizing f′(x) toprovide a domain pattern function f(x), wherein f(x) is a realconstant-amplitude approximation to f′(x).
 18. The method of claim 17,wherein said processing comprises low pass filtering, whereby a minimumdomain size in f(x) is increased compared to a minimum domain size ind(x).
 19. The method of claim 18, wherein said processing comprisesfiltering to reject quasi-phasematching harmonics.
 20. The method ofclaim 17, further comprising repeating said processing, said computingand said binarizing f′(x) in sequence until a termination condition issatisfied.
 21. The method of claim 20, further comprising monitoring aconvergence rate of said repeating and resetting said repeating if acondition of slow convergence is detected.
 22. The method of claim 8,further comprising: f) modifying said phase response φ(f) responsive toa figure of merit depending on A(f) and B(f).
 23. The method of claim22, further comprising g) iteratively repeating said steps (c) through(f) in sequence until a termination condition depending in part on saidfigure of merit is satisfied.
 24. A quasi-phasematched optical frequencyconverter comprising: a quasi-phasematched active region having a realconstant-amplitude domain pattern function d(x); wherein a function D(f)and the pattern function d(x) are a Fourier transform pair; wherein anon-negative real amplitude response A(f) and a real phase response φ(f)of the active region are defined via D(f)=A(f) exp(iφ(f)); whereind²φ(f)/df² is substantially proportional to A²(f).
 25. The opticalfrequency converter of claim 24, wherein a relation φ(f)∝∫₀ ^(f)(∫₀ ^(z)² A²(z₁)dz₁)dz₂ substantially holds between A(f) and φ(f).